Simplify the following expression and state the condition under which the simplification is valid. $p = \dfrac{n^3 - 7n^2 + 12n}{-5n^3 + 60n^2 - 135n}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ p = \dfrac {n(n^2 - 7n + 12)} {-5n(n^2 - 12n + 27)} $ $ p = -\dfrac{n}{5n} \cdot \dfrac{n^2 - 7n + 12}{n^2 - 12n + 27} $ Simplify: $ p = - \dfrac{1}{5} \cdot \dfrac{n^2 - 7n + 12}{n^2 - 12n + 27}$ Since we are dividing by $n$ , we must remember that $n \neq 0$ Next factor the numerator and denominator. $ p = - \dfrac{1}{5} \cdot \dfrac{(n - 3)(n - 4)}{(n - 3)(n - 9)}$ Assuming $n \neq 3$ , we can cancel the $n - 3$ $ p = - \dfrac{1}{5} \cdot \dfrac{n - 4}{n - 9}$ Therefore: $ p = \dfrac{ -n + 4 }{ 5(n - 9)}$, $n \neq 3$, $n \neq 0$